Cut faces where they intersect with the mesh
In Edit Mode (Use TAB to toggle), Loop select the front inner ring of edges of the barrel by placing your cursor over one of the edges that goes around the circle and hitting Alt+Shift.
Separate this from your mesh by hitting P >> S.
Use TAB to Toggle back into object mode.
RMB over the inner ring to select it.
Hold Shift and select the mesh that you need to cut.
Now go into front view Numpad 1.
Make sure that your toolshelf is open at the left - T to toggle.
Use TAB to toggle back into edit mode.
Select all (Verts / Faces).
On the toolshelf use "Knife Project" - There is a checkbox on the bottom left hand side of the toolshelf that says "Cut Through", make sure you check it.
Now just delete the edges and faces that you don't want.